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Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation

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Abstract

We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional (2D) domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (non-symmetric) ADI method. Moreover, we study the effect of the regularization by the isotropic background permeability \(r>0\), showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform the numerical error analysis of our method making use of the Wasserstein distance.

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Data Availability

All the data used in the numerical tests of this paper are defined in the main text, Section 4.

Notes

  1. The numerical scheme is implemented in Matlab and the solution of the linear system is computed with the “\” command.

  2. To calculate the Wasserstein distances between two vectors, we make use of the matlab function available at https://github.com/nklb/wasserstein-distance.

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Authors and Affiliations

  1. Applied Mathematics and Computational Sciences Department, King Abdullah University of Science and Technology, 4700, Thuwal, Saudi Arabia

    Clarissa Astuto, Daniele Boffi, Jan Haskovec & Peter Markowich

  2. Department of Mathematics “F. Casorati”, University of Pavia, 27100, Pavia, Italy

    Daniele Boffi

  3. Department of Mathematics, University of Vienna, 1090, Vienna, Austria

    Peter Markowich

  4. Department of Mathematics and Computer Science, University of Catania, 95125, Catania, Italy

    Giovanni Russo

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  1. Clarissa Astuto
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  3. Jan Haskovec
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  4. Peter Markowich
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  5. Giovanni Russo
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Contributions

PM and JH modeled the systems that govern this work. CA, DB, and GR set the methodologies and numerical schemes that were used. CA validated the results and wrote the original draft. CA, JH, and GR wrote, reviewed, and edited the manuscript. DB acquired funding and provided resources.

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Correspondence to Clarissa Astuto.

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Astuto, C., Boffi, D., Haskovec, J. et al. Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00297-3

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  • DOI: https://doi.org/10.1007/s42967-023-00297-3

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